SECT. III.] SPECIAL SERIES. 149 



It follows from this that the value of the finite series 

 sin x g sin 2# + ^ sin 3x - sin 5x -f- p sin 7x . . . -- sin mx 



differs very little from that of \x y when the number of terms 

 is very great ; and if this number is infinite, we have th,e known 

 equation 



^ x sin x ^ sin 2x + ^ sin 3x - 7 sin 4# -f -? sin 5# &c. 



Zi o 4&amp;lt; o 



From the last series, that which has been given above for 

 the value of JTT might also be derived. 



183. Let now 

 y = ^ cos 2x ^ cos 4x + - cos 6x - . . . 



COS ~ m -~- COS ~&quot; tx 



2m -2 



Differentiating, multiplying by 2 sin 2x } substituting the 

 differences of cosines, and reducing, we shall have 



ax cos x 



f, r, i 



or 



r j j r^ sm ( 2??i + 



= c - \dx tan x + \dx 2 



J J cosx 



integrating by parts the last term of the second member, and 



supposing 



equation 



y 



we suppose x nothing, we find 



supposing m infinite, we have y = c + log cos x. If in the 



y = ^ cos 2x - -r cos x + - cos Qx-- cos So; + . . . &c. 



Z T) o 



therefore y = - log 2 + 5 log cos ir. 



Thus we meet with the series given by Euler, 



log (2 cos #) = cos x - - cos 2# -f ^ cos 3x - -j cos 4^ + &c. 



